Generated Ideals and the Lattice of Ideals (Pt. I)
Point of Post: In this post we discuss the notion of generated ideals from subsets of a ring as well as the fact that the ideals of a ring form a complete modular lattice.
Much as was the case for normal subgroups of a group it’s true that there is some lattice like structure for ideals. In this post we shall describe the notion of generated ideals and use it to prove that the poset of ideals of is a complete modular lattice .
Intuitively what we’d like to figure out is if we have some ring and some subset of can we find some left, right, or two-sided ideal which contains and is ‘as small as possible’ (i.e. minimal with respect to containment). The first step in doing this is the following observation:
Theorem: Let be a ring and be a collection of ideals in . Then, is an ideal of .
Proof: We know that the intersection of subrings are subrings, and so it suffices to prove that has the two-sided absorption property. To see this let and be arbitrary. We note then that for every we have that and so . Thus, for all and so . Since was arbitrary we see that has the left absorption property. A similar idea shows that it has the right absorption property and so is an ideal as desired.
A similar proof shows that in fact the arbitrary intersection of right or left ideals in a ring is also a left or right ideal respectively.
With this in mind, if is a ring and we define the ideal generated by , denoted , to be equal to
Where denotes the set of all ideals of . We define the notions of the generated left ideal and generated right ideal by replacing with and respectively (where these denote the set of left and right ideals of ) and denote them and . If we denote as (similarly for and ). An ideal (left, right, or two-sided) is called finitely generated
So we have done what we’ve set out to do, namely show that any subset of a ring is contained inside some smallest ideal. That said, it isn’t very ‘nice looking’. As is the general case of math, once we’ve sucked the usefulness out of the abstract definition (in this case, it’s clear that our abstract desire of “minimality in terms of containment” is satisfied) we move on to a more practical definition, something we can actually work with (anyone having tried to do anything practical with singular homology understands this well). To do this though we need to make the concession that we are working with unital rings–but this really isn’t all too bad.
Theorem: Let be a unital ring and . Then, , , and where , and are as previously defined.
Proof: We prove only that since the other cases are done similarly. To do this we first note that evidently , since by definition every sum of the form where and is any ideal . So, if we can show that we’ll be done since it will be an ideal containing (since ) and so . So, to do this we let and be arbitrary, then and and thus, by the arbitrariness of and we know that has the left and right absorption properties, thus it remains to show that (as abelian groups). To see this we merely note that if and are in with then there difference is equal to where if and if , if and for , and finally for and for . But, this is clearly an element of and so . The conclusion follows from previous discussion.
Lattice of Ideals
With this notion of generated ideals in mind we can define the lattice structure on . Before we begin though we recall some of the basic concepts from order theory. Firstly, recall that if is a partially ordered set (poset) then a subset is said to have a supremum if there exist some such that for all and if is such that for all then . It’s easy to prove that if has a supremum it’s unique, and so we may unambiguously denote it by or (or even if ). Similarly, is said to have an infimum if there exists some such that for all and if is such that for all then . Once again, if has an infimum it’s unique and we may denote it or (or even if ).
A poset is called a complete lattice if and exist for all . What we now claim is that given any ring one has that is a complete lattice. Indeed:
Theorem: Let be a ring. Then is a complete lattice with and .
Proof: Let be any subset of . We have by previous theorem that , it’s clearly a lower bound for in , and evidently if is such that for all then and so is the infimum for . Similarly, we have by definition that contains and thus contains for each. Moreover, if contains for every then contains and thus . Thus, is the supremum of .
A similar statement holds for and with and in place of .
1. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.
2. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. Basic Abstract Algebra. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.